Modeling Processes of 4th-Year Middle-School Students and the Difficulties Encountered

Abstract

AbstractMathematics teachers have recently begun to stress the need for teaching models and modeling approaches that encompass cognitive and meta-cognitive thought processes for every level of schooling, starting from primary school through to higher education. The objective of this study is to examine modeling processes with the help of modeling activities for 4th-year middle-school students and to determine the difficulties encountered in the processes. The study was conducted with 4th-year students from a lower socioeconomic status who were enrolled in a public middle school. A preliminary study was carried out. Later, three students were chosen as a focus group to work on a model-eliciting activity called the Volleyball Problem, and the entire process was recorded on video. The mathematical thoughts that the students developed during the modeling process and their written responses were then qualitatively analyzed. The results obtained showed that the students were able to produce many new and different ideas. They were able to discuss various assumptions before reaching a conclusion and the activities helped them to think deeper and develop their mathematical thinking. At the same time, however, it was found that the students experienced certain difficulties during the model-eliciting process in terms of understanding, developing, and constructing an adequate model.Keywords: Middle-school students * Model-eliciting activities * Mathematical modeling * Difficulties or challengesIn recent years, one of the basic concerns that have led mathematics teachers to work on approaches to models and modeling has been the inadequacy of traditional problem-solving activities in teaching students how to develop their problem-solving skills (Greer, 1997; Mousoulides, Christou, & Sriraman, 2006; Schoenfeld, 1992 as cited in Kertil, 2008). The importance of this lies in the fact that one of the most prominent goals of education today is to train individuals who are capable of analytical thinking and have the capacity to generate effective and creative solutions for problems (Kilpatrick, 1992). Educators in different fields emphasize that success after school depends upon new understanding and skills. These skills include the abilities of structuring, defining, clarifying, manipulating, and projecting within complex systems (English, 2002). It is these skills that play an important role in achieving success, planning multidimensional and multifaceted projects, and controlling and developing communications (Gainsburg, 2006). In addition, these skills contribute greatly to interpreting and mentally analyzing conceptual structures (English, 2002; Gainsburg, 2006; Lesh & Doerr, 2003). The approach that will enable students to develop such skills as engaging model-eliciting activities that lead to mathematical modeling (English & Watters, 2005; Lesh & Doerr, 2003).In its most general sense, mathematical modeling is the mathematical expression of real life experience. This is, however, more than taking a situation from the real world and interpreting it based on simple formulations using appropriate variables. Modeling encompasses the processes of observing the situation, identifying the relationships involved, applying mathematical analysis, developing model,s and revisiting the interpretations of these (Swetz & Hartzler, 1991). It was the applicability of mathematics to real life situations, the facility with which mathematical knowledge lends itself to concrete application, and the way mathematics can be used to produce more analytical and practical solutions that led to the birth of the idea that mathematical modeling should be taught in primary and middle schools (Mousoulides et al., 2006).Studies conducted in primary schools have shown that modeling activities provide students with: (a) a powerful tool to use in developing critical and high-level thinking skills (English & Watters, 2005), (b) a new and effective learning environment where existing deficiencies in conceptual knowledge are identified and new mathematical knowledge is gained (Chamberlin, 2004), (c) encouragement to use different and various representations to explain conceptual systems (Boaler, 2001; English & Watters, 2004; Mousoulides, 2007), and (d) increased communication skills by encouraging students to share their own mathematical notions and understanding (English, 2006). …